An introduction to the general properties of the Fourier series
Introduction
In this module we will discuss the basic properties of the Continuous-Time Fourier Series. We will begin by refreshing your memory of our basic
Fourier series equations:
Let
$\mathcal{F}(\xb7)$ denote the transformation from
$f(t)$ to the Fourier coefficients
$$\mathcal{F}(f(t))=\forall n, n\in \mathbb{Z}\colon {c}_{n}$$$\mathcal{F}(\xb7)$ maps complex valued functions to sequences of
complex numbers .
Linearity
$\mathcal{F}(\xb7)$ is a
linear transformation .
If
$\mathcal{F}(f(t))={c}_{n}$ and
$\mathcal{F}(g(t))={d}_{n}$ .
Then
$$\forall \alpha , \alpha \in \mathbb{C}\colon \mathcal{F}(\alpha f(t))=\alpha {c}_{n}$$ and
$$\mathcal{F}(f(t)+g(t))={c}_{n}+{d}_{n}$$
Easy. Just linearity of integral.
$\mathcal{F}(f(t)+g(t))=\forall n, n\in \mathbb{Z}\colon \int_{0}^{T} (f(t)+g(t))e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega}_{0}nt)}\,d t+\frac{1}{T}\int_{0}^{T} g(t)e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon {c}_{n}+{d}_{n}={c}_{n}+{d}_{n}$
$\mathcal{F}(f(t-{t}_{0}))=e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}$ if
${c}_{n}=\left|{c}_{n}\right|e^{i\angle ({c}_{n})}$ ,
then
$$\left|e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}\right|=\left|e^{-(i{\omega}_{0}n{t}_{0})}\right|\left|{c}_{n}\right|=\left|{c}_{n}\right|$$$$\angle (e^{-(i{\omega}_{0}{t}_{0}n)})=\angle ({c}_{n})-{\omega}_{0}{t}_{0}n$$
$\mathcal{F}(f(t-{t}_{0}))=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t-{t}_{0})e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(t-{t}_{0})e^{-(i{\omega}_{0}n(t-{t}_{0}))}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(\stackrel{~}{t}())e^{-(i{\omega}_{0}n\stackrel{~}{t})}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon e^{-(i{\omega}_{0}n\stackrel{~}{t})}{c}_{n}$
A differentiator
attenuates the low
frequencies in
$f(t)$ and
accentuates the high frequencies. It
removes general trends and accentuates areas of sharpvariation.
A common way to mathematically measure the smoothness of a
function
$f(t)$ is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of thesignal, specifically how fast they
decay as
$n\to $∞ .If
$\mathcal{F}(f(t))={c}_{n}$ and
$\left|{c}_{n}\right|$ has the form
$\frac{1}{n^{k}}$ ,
then
$\mathcal{F}(\frac{d^{m}f(t)}{dt^{m}})=(in{\omega}_{0})^{m}{c}_{n}$ and has the form
$\frac{n^{m}}{n^{k}}$ .So for the
${m}^{\mathrm{th}}$ derivative to have finite energy, we need
$$\sum \left|\frac{n^{m}}{n^{k}}\right|^{2}$$∞ thus
$\frac{n^{m}}{n^{k}}$ decays
faster than
$\frac{1}{n}$ which implies that
$$2k-2m> 1$$ or
$$k> \frac{2m+1}{2}$$ Thus the decay rate of the Fourier series dictates
smoothness.
Fourier differentiation demonstration
Integration in the fourier domain
If
$\mathcal{F}(f(t))={c}_{n}$
then
$\mathcal{F}(\int_{()} \,d \tau )$∞tfτ1ω0ncn
If
${c}_{0}\neq 0$ , this expression doesn't make sense.
Integration accentuates low frequencies and attenuates high
frequencies. Integrators bring out the
general
trends in signals and suppress short term variation
(which is noise in many cases). Integrators are
much nicer than differentiators.
Fourier integration demonstration
Signal multiplication and convolution
Given a signal
$f(t)$ with Fourier coefficients
${c}_{n}$ and a signal
$g(t)$ with Fourier coefficients
${d}_{n}$ ,
we can define a new signal,
$y(t)$ ,
where
$y(t)=f(t)g(t)$ .
We find that the Fourier Series representation of
$y(t)$ ,
${e}_{n}$ ,
is such that
${e}_{n}=\sum_{k=()} $∞∞ckdn-k .
This is to say that signal multiplication in the time domainis equivalent to
signal convolution in the frequency domain, and vice-versa: signal multiplication in the frequency domain is equivalent to signal convolution in the time domain.The proof of this is as follows
Like other Fourier transforms, the CTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.
Properties of the ctfs
Property
Signal
CTFS
Linearity
$ax\left(t\right)+by\left(t\right)$
$aX\left(f\right)+bY\left(f\right)$
Time Shifting
$x(t-\tau )$
$X\left(f\right){e}^{-j2\pi f\tau /T}$
Time Modulation
$x\left(t\right){e}^{j2\pi f\tau /T}$
$X(f-k)$
Multiplication
$x\left(t\right)y\left(t\right)$
$X\left(f\right)*Y\left(f\right)$
Continuous Convolution
$x\left(t\right)*y\left(t\right)$
$X\left(f\right)Y\left(f\right)$
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?